Arf invariant of a knot

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In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If F is a Seifert surface of a knot, then the homology group H1(FZ/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.

Definition by Seifert matrix[edit]

Let be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surface of genus g which represent a basis for the first homology of the surface. This means that V is a 2g × 2g matrix with the property that V − VT is a symplectic matrix. The Arf invariant of the knot is the residue of

Specifically, if , is a symplectic basis for the intersection form on the Seifert surface, then

where denotes the positive pushoff of a.

Definition by pass equivalence[edit]

This approach to the Arf invariant is due to Louis Kauffman.

We define two knots to be pass equivalent if they are related by a finite sequence of pass-moves,[1] which are illustrated below: (no figure right now)

Every knot is pass-equivalent to either the unknot or the trefoil; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.[2]

Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.

Definition by partition function[edit]

Vaughan Jones showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a knot diagram.

Definition by Alexander polynomial[edit]

This approach to the Arf invariant is by Raymond Robertello.[3] Let

be the Alexander polynomial of the knot. Then the Arf invariant is the residue of

modulo 2, where r = 0 for n odd, and r = 1 for n even.

Kunio Murasugi[4] proved that the Arf invariant is zero if and only if Δ(−1)  ±1 modulo 8.

Notes[edit]

  1. ^ Kauffman (1987) p.74
  2. ^ Kauffman (1987) pp.75–78
  3. ^ Robertello, Raymond, Communications on Pure and Applied Mathematics, Volume 18, pp. 543–555, 1965
  4. ^ Murasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69–72

References[edit]